/* Reconstructing a Motor from Exact Point Correspondences according to Sect. 6.8 of the tutorial A Guided Tour to the Plane-Based Geometric Algebra PGA by Leo Dorst, University of Amsterdam */ ExpApprox = {1 + _P(1) + _P(1)*_P(1)/2 + _P(1)*_P(1)*_P(1)/6 +_P(1)*_P(1)*_P(1)*_P(1)/24} Motor = { // computes the motor between two points, lines or planes // as the sqrt of them !temp = 1+_P(1)/_P(2); !abstemp = abs(temp); temp/abstemp } // original points A1 = createPoint(ax, ay, az); B1 = createPoint(bx, by, bz); C1 = createPoint(cx, cy, cz); // arbitrary transformation !GT = ExpApprox(0.3*(e1^e3) + 0.2*(e2^e0)); // corresponding target points ?At = GT * A1 * ~GT; ?Bt = GT * B1 * ~GT; ?Ct = GT * C1 * ~GT; // Transformation from A1 to At // (translation) !VA = Motor(At, A1); !A2 = VA * A1 * ~VA; !B2 = VA * B1 * ~VA; !C2 = VA * C1 * ~VA; // Transformation from B2 to Bt // based on the rotation from the line L2 to L1 !L1 = *(*At ^ *Bt); !L2 = *(*At ^ *B2); !VB = Motor(L1, L2); !B3 = VB * B2 * ~VB; !C3 = VB * C2 * ~VB; // Transformation from C3 to Ct // based on the rotation of two planes !P1 = *(*L1 ^*Ct); !P2 = *(*L1 ^*C3); !VC = Motor(P1,P2); // complete transformation !V = VC * VB * VA; // interplation motor lerp = 1 * (1-t) + V * t; // interpolated points ?AI = lerp * A1 * ~lerp; ?BI = lerp * B1 * ~lerp; ?CI = lerp * C1 * ~lerp;